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– STATISTICS
HISTOGRAMS, GRAPHS, MEAN, MEDIAN
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HISTOGRAMS, GRAPHS, MEAN, MEDIAN
- In general, can you recreate the original data values from its histogram? a. Yes b. No
- We know that if a histogram is skewed left, then the mean is less than the median. WHY? a. There are a few small values in the data set compared to the rest. b. There are a few large values in the data set compared to the rest.
- Suppose an exam is 48 minutes long and most of the students stayed until the very end although there were a few students who finished early. You collect data on how much time each student spent taking the exam and make a histogram. The shape of the histogram will be what? (Hint: DRAW A PICTURE) a. Skewed left b. Skewed right c. Symmetric
- Suppose you have a numerical data set that is very much skewed to the left. If you had to pick one, which measure of center best represents most of the data in this case? WHY? a. Mean b. Median
- It is possible for the average of a data set to be larger than most of the values in that data set. a True b False
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- The mean of a data set must be one of the values in the data set. a. YES b. NO
- The median of a data set must be one of the values in the data set.
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Frequency
Histogramof Withdraw 30
25
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5
(^0100 200 ) Withdraw
400 500
Next 2 problems: Withdrawals from a bank are recorded over a day’s time and summarized in the histogram below.
- What is a better name for the Y-axis here? a. Number of Withdrawals b. Count c. Monday
- Two measures of center were calculated: 176 and 160. One of these numbers is the mean, and the other is the median. Which one is the median? a. 176 is the median b. 160 is the median c. Can’t tell
- Last year a small accounting firm paid each of its five clerks $25,000; two junior accountants received $60,000 each; and the firm’s owner got paid $255,000. This year the
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firm gave no raises to the clerks and the junior accountants, but the owner’s salary is increased to $455,000. How does this year’s raise affect the average salary for the accounting firm?
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new propane fuel system. He takes a random sample of taxis from his fleet, and calculates the miles per gallon for each taxi on propane. The results are shown below.
Descriptive Statistics: Mileage
- Why would you say the data must be close to symmetric? a. Because it is easier to interpret symmetric data b. Because the mean and median are close to each other c. Because the minimum and maximum are close together d. There is not enough information to determine that the data are close to symmetric
- Suppose after taking the sample, the owner notices that the next taxi in his fleet gets 35 miles per gallon. Which of the following statistics will be the least affected by adding this taxi to the data set? a. The mean b. The median c. The maximum d. All of these will be affected the same amount
- Which of the following data sets has more variability around the mean, the first or the second?
A. 40 30 20 10
Variable Count Mean StDev Minimum Q1 Median Q3 Maximum Mileage 50 19.784 3.093 13.300 17.800 20.000 21.825 26.
Frequency
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B.
Next 3 questions: refer to the histogram below, describing the amount of time required to answer questions on a telephone support line. There were 300 calls altogether.
- The number of calls that last more than 18 minutes is approximately: a. 10 b. 30
40 30 20 10 (^020 40 60 80 )
Frequency
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- The shape of the data set is a. skewed right. b. skewed left. c. symmetric.
- The median of this dataset is: (choose best answer) a. 18 b. 5 c. 12 d. 14
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- The Milbert Marketing Group recently conducted a study of buying habits of the residents of Apex, North Carolina. From the Apex telephone directory they randomly selected 200 individuals and asked them how much they spent on purchases of DVD movies in the past month. Suppose we know that the majority of the residents did not purchase any DVDs in the past month. This indicates a The median amount spent on DVDs would be greater than $32. b The median amount spent on DVDs would be less than $32. c The median amount spent on DVDs would be $32.
- The average score for a class of 30 students was 75. The 20 male students in the class averaged 70. The 10 female students in the class averaged: a. 75 b. 85 c. 65
Next 6 questions: Data are collected on the number of sales made in a given week by each sales person in a certain company. The same data set is displayed in each of the three histograms below. Compare and contrast these histograms and answer the questions below.
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- What does the “X” variable stand for on the horizontal axis?
Histogram# 25 20
15 10 5
(^10 20 30) X 40 50
frequency
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The number of sales made in a given week for a salesperson.
- Write down a better word than “Frequency” to describe the Y-axis. Number of sales personnel.
- How many sales personnel are there in this company? Each observation represents the sales of a salesperson, so we can just add the frequencies together. So there are about 12+15+9+15+8 = 59 salespeople working for this company.
- Which histogram makes the differences in the sales appear to be the most dramatic? Explain what was done to make it look this way. The second histogram, since the columns look taller. The Y axis scale was made taller.
- Which histogram makes the differences in the sales appear to be the least dramatic? Explain what was done to make it look this way. The third, since the columns look shorter. The Y axis scale was made shorter.
- Which graph do you think is the most appropriate display? Justify your answer. This is subjective, but the first one is a good compromise and is easy to read.
THE CURRENT PRICE OF GAS
Gasbuddy.com is a website where people report gas prices in their areas all over the country (and Canada) and all kinds of statistics are collected and reported. This website contains thousands of gas prices. Go to their website at http://www.gasbuddy.com/ and use it to break things down. Note: If you get lost anywhere, retype Gasbuddy.com – at times I have found the pages direct you to other website names.
- On the home page, the very first table you see shows people reporting gas prices around the area. Put in the OSU zip code: 43210, and write down 5 of the gas prices you see. Do they reflect what you think is the current gas price around OSU? How credible is this data?
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HISTOGRAMS, GRAPHS, MEAN, MEDIAN
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There is bias in this process – explain. (Use your notes to find the answers.) There is volunteer bias (or self-selection bias) in this survey.
- While the data does give a rough idea of what’s going on with gas prices in our area, there is bias in this data due to other items. Describe 2 ways the data are going to be biased OTHER than the kind you identified in #2 above (you do not need to name the type of bias- some biases don’t have names.) Any reasonable answer is OK. Some possibilities are undercoverage (people without smart phones are less likely to provide information) as well as sampling bias (since
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we have no way of guaranteeing the sample is random – that is, each person in the population does not have an equal chance of providing information.
- On the 2nd^ table on the home page you see “ Columbus Stats” where they show the average price, lowest price, and the current trend. What data are they using to calculate the average price and the current trend? If you can’t find an explanation on the website, how do you think they could/should be doing it? Any reasonable answer is OK. The website is not explicit in telling us how they calculate the average price. One reasonable way to calculate the average would be to average all the prices for an area for the last 24 hours. One reasonable way to calculate the trend would be to look at the average gas prices over the last three days.
- There is no measure of variation mentioned in any of their statistics. Explain why this is a problem. Without the standard deviation, there is no way of understanding how the gas prices might vary throughout the area.
- Near the bottom of the home page you see “ FEATURES” and one of them is the “Gas Price Heat Map”, which gives you different colors to indicate high or low gas prices across the U.S. Name two confounding variables that that affect gas prices that need to be taken into account. Answers will vary. Some options are: local taxes, distance to gas supplies (as transportation costs affect gas prices), other local regulations, etc.
- Back to home page : Also near the bottom of the home page right below FEATURES , you see a section called “ STATS”. Go to the Gas Price Charts and see their US average gas prices charted over time. At each time point they have calculated the average gas price. Discuss how difficult it would be to collect data that truly represents the U.S in terms of gas price at one period in time.
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equally likely.
- Using this same Gas Price Chart , look at the trend in US reported average gas prices over time. Describe the trend. Gas prices across the US have trended downward in recent history.
- Now look at how the U.S. trend compares with Columbus OH, by choosing Columbus from the “ Area 2 ” drop down menu of the Gas Price Chart. Describe what you see.
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Both have trended downward over time. The Columbus, OH average appears to follow the national average pretty closely, though at times it is pretty far below the national average.
10. Now look at how the trend compares with Los Angeles, CA. What do you see and why? The national average has been much lower than the average gas price in Los Angeles. 11. How does a large city like Los Angeles CA affect the overall mean gas price in the U.S? It raises the average, since there are a lot of gas stations in Los Angeles with higher gas prices.
- Check out trends for other states compared to the U.S. and describe any interesting findings. Any reasonable answers are OK. Canada has had a higher average gas price. Houston, TX has had a lower average gas price. 13. From this same page, click on “Media” way at the top. These are statistics that GasBuddy provides for the media to use when talking about gas prices. Click on the gas pump icon near the top that says “Click here to see more price statistics for the nation” A table of statistics should pop up. NOTE: If you get lost, go back and type in Gasbuddy.com and click on Media way at the top. a. You see the median gas price is reported. Why does this help, compared to the mean? The median always tells you the number where 50% of the data are higher and 50% are lower. This is an informative number, since the data are potentially skewed by areas (such as Los Angeles) that have such a higher gas price.
b. You see other statistics reported such as the lowest 5% mark. Describe in general what these statistics are telling you about current gas prices. Which ones are most
